1. Field of the Invention
The present invention relates to data analysis by determining symmetry and related parameters within a set of N points, particularly in image analysis and fault detection applications.
2. Description of Related Art
Prior methods handle symmetry on a piece-meal basis, working separately with translational, mirror or rotational symmetry through the localizations of symmetry axes or centers, which must be calculated separately for each case. Mathematically, this procedure can lead to the complicated construction of various symmetry groups as outlined by Jablan (1995). Attempts to work with partial symmetry have led to even more complicated mathematical structures such as groupoids (cf. Weinstein and Johnson).
In a previous study by the inventor involving minimum laterally-adiabatically reduced Fisher information (see Collins 2005) which led to the development of the present method, the definition of “tropical” symmetric order is given as the “number of equal distances from each object, added together”. It was further determined that working with reference distances could be avoided by counting numbers of pairs of equal distance, since one reference distance plus one equal distance counts as one pair of equal distances.
Alexander (2002) counts the number of “local symmetries” (pg. 189); however, this method appears to be infeasible for large N (taking exponential complexity) since it involves considering all subsets, plus there is no indication of how to count local symmetries in more than one dimension.
In the work of Kia, the term “distance entropy” occurs with respect to the “distance transform”. This term differs from its use in the present invention since Kia refers to calculations involving edge effects.
As stated in the article by Reid (2006), most imaging problems can be solved at present by “proper knowledge, hardware, software, and tweaking . . . ”. The present invention attempts to solve the problem of measuring symmetry of different types within a single data set simultaneously and allowing interchangeable symmetry calculations.